I can only go to Object Mode
There are mainly two ways to switch between Object and Edit mode.

1. TAB


As @depperm suggests you need to have at least one object selected;
you also need to have the mouse pointer inside the 3D View window;
just press TAB to switch.


2. Mode menu


In the 3D View header you'll find a mode menu (see picture below);

just click on it and select the mode you want to switch to.